Of course, the general process for constructing \(\hat{X}\)  is non
constructive, as it involves the axiom of choice to turn \({A}^f\) 
into an almost disjointed generating family.
This is why, in the last two sections of this paper, we explore a
particular case where we are able to describe explicitly \(\hat{X}\)  as
a quotient of a fiber bundle, all “hands on”.
We start with the measure space \((\mathbb {R}^m,{B}(\mathbb {R}^m),{I}^k_\infty )\) 
where \(1 \leqslant k \leqslant m-1\)  are integers, \({B}(\mathbb {R}^m)\)  is the
\(\sigma \) -algebra of Borel subsets of \(\mathbb {R}^m\) , and \({I}^k_\infty \)  is
the integral geometric measure described in [1]} and
[2]}.
Note that it is not semi-finite, [1]}.
Thus, we replace it with its complete semi-finite version
\((\mathbb {R}^m,\widetilde{{B}(\mathbb {R}^m)},\tilde{{I}}^k_\infty )\) .
We let \({E}\)  be the collection of \(k\) -dimensional submanifolds \(M\subseteq \mathbb {R}^m\)  of class \(C^1\)  such that \(\phi _M = {H}^k \hspace{2.5pt}{\vrule height7pt width.5pt depth0pt}\hspace{-0.2pt}\vbox {\hrule height.5pt width7pt depth0pt}\, M\)  is
locally finite.
It follows from the Besicovitch Structure Theorem [1]}
that \({E}\)  is \({N}_{\tilde{{I}}^k_\infty }\) -generating,
REF (ii).
Now, for each \(x \in \mathbb {R}^m\)  we define \({E}_x = {E}\cap \lbrace  M : x \in M \rbrace \)  and we define on \({E}_x\)  an equivalence relation as follows.
We declare that \(M \sim _x M^{\prime }\)  if and only if
\(\lim _{r \rightarrow 0^+} \frac{{H}^k(M \cap M^{\prime } \cap \mathbf {B}(x,r))}{\alpha (k)r^k}= 1.\) 
